Real zeros of random cosine polynomials with palindromic blocks of coefficients

TL;DR

This paper investigates how palindromic block structures in coefficients of random cosine polynomials affect the expected number of real zeros, showing an increase compared to the classical i.i.d. case.

Contribution

It introduces a new model of dependent coefficients with palindromic blocks and derives explicit asymptotics for the expected number of real roots, revealing an increase over the classical i.i.d. case.

Findings

Expected real zeros are asymptotically proportional to 2n/√3 with a constant greater than 1.

The constant depends on block length and is given by a double integral formula.

Polynomials with palindromic blocks have more real zeros than classical i.i.d. coefficient polynomials.

Abstract

It is well known that a random cosine polynomial $ V_n(x) = \sum_ {j=0} ^{n} a_j \cos (j x) , \ x \in (0,2 \pi) $, with the coefficients being independent and identically distributed (i.i.d.) real-valued standard Gaussian random variables (asymptotically) has $ 2n / \sqrt{3} $ expected real roots. On the other hand, out of many ways to construct a dependent random polynomial, one is to force the coefficients to be palindromic. Hence, it makes sense to ask how many real zeros a random cosine polynomial (of degree $ n $) with identically and normally distributed coefficients possesses if the coefficients are sorted in palindromic blocks of a fixed length $ \ell. $ In this paper, we show that the asymptotics of the expected number of real roots of such a polynomial is $ \mathrm{K}_\ell \cdot 2n / \sqrt{3} $, where the constant $ \mathrm{K}_\ell $ (depending only on $ \ell $) is greater than 1, and can be explicitly represented by a double integral formula. That is to say, such polynomials have slightly more expected real zeros compared with the classical case with i.i.d. coefficients.